The equation of the line which passes through the point (-3,8) and cuts of +ve intercepts on the axes whose sum is 7 is

Options

A. `3x-4y=12`
B. `4x+3y=12`
C. `3x+4y=12`
D. `4x-3y=12`

Answer

To find the equation of the line that passes through the point (-3, 8) and cuts off positive intercepts on the axes whose sum is 7, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Intercept Form of the Line**:
The equation of a line in intercept form is given by:
\[ \frac{x}{A} + \frac{y}{B} = 1 \]
where \(A\) is the x-intercept and \(B\) is the y-intercept.

2. **Set Up the Conditions**:
We know from the problem that:
- The line passes through the point (-3, 8).
- The sum of the positive intercepts \(A + B = 7\).

3. **Substituting the Point into the Line Equation**:
Since the line passes through (-3, 8), we substitute \(x = -3\) and \(y = 8\) into the intercept form:
\[ \frac{-3}{A} + \frac{8}{B} = 1 \]
This gives us our first equation (let's call it Equation 1).

4. **Express \(B\) in Terms of \(A\)**:
From the condition \(A + B = 7\), we can express \(B\) as:
\[ B = 7 - A \]
Substitute this into Equation 1.

5. **Substituting \(B\) into Equation 1**:
Replace \(B\) in Equation 1:
\[ \frac{-3}{A} + \frac{8}{7 - A} = 1 \]

6. **Clearing the Denominator**:
To eliminate the fractions, multiply through by \(A(7 - A)\):
\[ -3(7 - A) + 8A = A(7 - A) \]
Expanding this gives:
\[ -21 + 3A + 8A = 7A - A^2 \]

7. **Rearranging the Equation**:
Combine like terms:
\[ -21 + 11A = 7A - A^2 \]
Rearranging gives:
\[ A^2 + 4A - 21 = 0 \]

8. **Solving the Quadratic Equation**:
Now we can solve the quadratic equation \(A^2 + 4A - 21 = 0\) using the quadratic formula:
\[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 4\), and \(c = -21\):
\[ A = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1} \]
\[ A = \frac{-4 \pm \sqrt{16 + 84}}{2} \]
\[ A = \frac{-4 \pm \sqrt{100}}{2} \]
\[ A = \frac{-4 \pm 10}{2} \]
This gives us two possible values for \(A\):
\[ A = 3 \quad \text{or} \quad A = -7 \]

9. **Finding Corresponding \(B\)**:
Using \(B = 7 - A\):
- If \(A = 3\), then \(B = 7 - 3 = 4\).
- If \(A = -7\), then \(B = 7 - (-7) = 14\) (not valid as we need positive intercepts).

10. **Final Equation of the Line**:
Using \(A = 3\) and \(B = 4\), the equation of the line is:
\[ \frac{x}{3} + \frac{y}{4} = 1 \]
Multiplying through by 12 to eliminate the fractions:
\[ 4x + 3y = 12 \]

### Conclusion:
The equation of the line is:
\[ 4x + 3y = 12 \]